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随机面积：正方形 vs 矩形

Random Areas

专题
Probability / 概率
难度
L4

题目详情

概率题：随机面积：正方形 vs 矩形。

英文原题

Peter and Paula both want to cut out a rectangular piece of paper. Because they are both probabilists they determine the exact form of the rectangle by using realizations of a positive rv, say U, as follows. Peter is lazy and generates just a single realization of this rv; he then cuts out a square that has length and width equal to this value. Paula likes diversity and generates two independent realizations of U. She then cuts out a rectangle with width equal to the first realization and length equal to the second realization.

a. Will the areas cut out by Peter and Paula differ in expectation?

b. If they do, is Peter's or Paula's rectangle expected to be larger?

解析

Peter 面积为 U2U^2，Paula 面积为 U1U2U_1U_2（独立同分布）。\n\n\nmathbbE[U1U2]=mathbbE[U]2,\n\n\\mathbb{E}[U_1U_2]=\\mathbb{E}[U]^2,\n\n\n\nmathbbE[U2]=mathbbE[U]2+mathrmVar(U)gemathbbE[U]2.\n\n\\mathbb{E}[U^2]=\\mathbb{E}[U]^2+\\mathrm{Var}(U)\\ge \\mathbb{E}[U]^2.\n\n\n因此 Peter 的期望面积更大，差值为 mathrmVar(U)\\mathrm{Var}(U)。

英文解析

Peter just generates a single realization of U, and so the area of his square will be equal to U2. Thus, on average the area of his square is equal to APeter = E[U2]. Paula generates two independent realizations of U for the width and length of her rectangle, let us call these realizations U1 and U2. Accordingly, the area of her rectangle will be equal to U1 - U2. On average the area of her rectangle equals

APaula=E[U1⋅U2]=E[U1]⋅E[U2]=E2[U]\begin{array}{r l} & {A_{P a u l a} = \mathsf{E}[\mathbf{U}_{1}\cdot \mathbf{U}_{2}]}\\ & {\qquad = \mathsf{E}[\mathbf{U}_{1}]\cdot \mathsf{E}[\mathbf{U}_{2}]}\\ & {\qquad = \mathsf{E}^{2}[\mathbf{U}]} \end{array}

because U1 and U2 are independent and have the same distribution as U. Therefore,

APeter−APaula=E[U2]−E2[U]=Var[U]≥0A_{P e t e r} - A_{P a u l a} = \mathsf{E}[\mathbf{U}^{2}] - \mathsf{E}^{2}[\mathbf{U}] = \mathsf{V a r}[\mathbf{U}]\geq 0

that is, APeter≥APaulaA_{Peter} \geq A_{Paula} . On average, the area of Peter's square is larger than that of Paula's rectangle, even though all lengths and widths of all rectangles (a square is a rectangle) are generated by realizations of the same generic rv, U.